F. Classical Relativistic Spin-Theory and Its Quantization

John French

Malcolm H. Mac Gregor

Heinz Krüger

Kenneth G. Dyall, Knut Fægri

The purpose of this chapter is to introduce the Dirac equation, which will provide us with a basis for developing the relativistic quantum mechanics of electronic systems. Thus far we have reviewed some basic features of the classical relativistic theory, which is the foundation of relativistic quantum theory. As in the nonrelativistic case, quantum mechanical equations may be obtained from the classical relativistic particle equations by use of the correspondence principle, where we replace classical variables by operators. Of particular interest are the substitutions In terms of the momentum four-vector introduced earlier, this yields In going from a classical relativistic description to relativistic quantum mechanics, we require that the equations obtained are invariant under Lorentz transformations. Other basic requirements, such as gauge invariance, must also apply to the equations of relativistic quantum mechanics. We start this chapter by reexamining the quantization of the nonrelativistic Hamiltonian and draw out some features that will be useful in the quantization of the relativistic Hamiltonian. We then turn to the Dirac equation and sketch its derivation. We discuss some properties of the equation and its solutions, and show how going to the nonrelativistic limit reduces it to a Schrödinger-type equation containing spin.

Yu. M. Shirokov

H. J. Briegel, Berthold‐Georg Englert, Georg Süssmann

Abstract When quantizing the classical Hamiltonian H (q,p,t) = [m2c4 + (c p - e A(t,q))2]1/2 + e V(t, q), commutators of the form [f (a), g(b)], where [a,b] is not a c-number, have to be evaluated. The concept of continuously symmetrized products enables us to derive a number of statements, such as a continuity equation for the density operator δ{q(t)-x)=n(t, x), in a formally concise way. We can also show, then, that the dependence of the Hamiltonian on higher powers of the kinetic momentum destroys the relativistic invariance of the theory, even if we admit a more general coupling of the external potentials than above.